Zhc ITlniverstt? of Cbicago 



The Analysis of Mental Functions 



A DISSERTATION 

submitted to the faculty of the 

Graduate School of Art and Literature 

in candidacy for the degree of 

doctor of philosophy 

(DEPARTMENT OF psychology) 



BY 



CURT ROSENOW 



A Private Edition Distributed by The University of Chicago Libraries 



A Trade Edition is Published by 

The Psychological Review Company, Princeton, N.J. 

As Psychological Monograph No. 000 



Gbe iftnivereiti? of Chicago 



The Analysis of Mental Functions 



A DISSERTATION 

submitted to the faculty of the 

Graduate School of Art and Literature 

in candidacy for the degree of 

doctor of philosophy 

(department of psychology) 



BY 

CURT ROSENOW 



A Private Edition Distributed by The University of Chicago Libraries 

A Trade Edition is Published by 

The Psychological Review Company, Princeton, N. J., 

As Psychological Monograph No. 000 






71^3 L .. - ■ 

&fe" 4 : . " 7 




A graceful custom offers me the welcome opportunity to give 
expression to a part of the sense of indebtedness which I feel. 
If this little work be found to have any merit at all, I believe that 
merit to lie largely in the point of view from which it is written. 
There is a sense in which that point of view is my own. There 
is another sense in which it is entirely due to the minds with 
which I have had contact. In that sense, it is my privilege to 
express my obligation to William James, whom I was not for- 
tunate enough to know in person, to James R. Angell, to George 
Herbert Mead, and to Addison W. Moore. Many others have 
given their grateful testimony with regard to the spirit of scien- 
tific comradeship which prevails in the department over which 
Professor Angell presides. I wish to add to this my apprecia- 
tion of his rare gift of appreciating and encouraging many di- 
vers and divergent types of mind. I feel that he who cannot 
work out his intellectual and psychological salvation here, can 
do so nowhere. 

My thanks are due to Prof. Harvey A. Carr for thorough and 
stimulating instruction. I wish to say, furthermore, that my in- 
terest in the subject of modern statistical method is due to my 
contact with the enthusiasm and the ability of Dr. Beardsley 
Ruml. And I wish to voice my gratitude to Dr. H. D. Kitson 
whose kindness and scientific courtesy in furnishing me with 
data has made this little study possible. 

Curt Rosenow. 
University of Chicago. 
June 8, 191 7 



I. The Problem. 

Whipple, in the introduction to his Manual, tells us that a 
"mental test" is the experimental determination for a given in- 
dividual of some phase of his mental capacity, the scientific 
measurement of some one of his mental traits. He goes on to 
say that its purpose is practical and diagnostic rather than theo- 
retical and analytic, but he recognizes that theory and analysis 
are likely to interact with practice and diagnosis in a way which 
is beneficial to both. Then, with a candor which cannot be 
praised too highly, he tells us that there is as yet no such thing as 
a science of mental tests. "... there is, at the present time, 
scarcely a single mental test that can be applied unequivocally as 

a psychical measuring rod we too often do not know 

what we are measuring; and we too seldom realize the astound- 
ing complexity, variety and delicacy of form of our psychical 
nature." And finally we are told that the pressing need of the 
day is not the inventing of new tests, but the exhaustive investi- 
gation of those we already have. 

I cannot subscribe too heartily to some of these sentiments. 
To be sure, Whipple thinks that rigid standardization and the 
setting up of norms is the most urgent need, whereas I believe 
that new tests of the right kind and evaluation of old tests, is 
what the situation calls for. It is all too true that we do not 
know what we are measuring, and it seems a rather futile pastime 
to standardize the measures of we know not what. We need to 
devise tests so that we do know what we are measuring, and in 
order to do so we must subject our so-called tests to intensive 
study and analysis. Only so can standardization proceed intel- 
ligently. 

What is it that we are measuring? In the first place, and it 
seems to me that some of our mental testers have quite lost sight 
of this obvious fact, we are measuring actual, factual perform- 
ance at some definite specific task, like thinking of the "opposites" 
of a list of adjectives, or recalling the names of a number of 



2 CURT ROSEN OW 

familiar objects which have been seen during a brief interval of 
time. In the second place we postulate, assume, or fondly hope 
that this actual performance is symptomatic of ability in some 
wider range of mental ability which we call a mental function. 1 
We hope that excellence at the "opposites" test is symptomatic 
of ability quickly to recall the appropriate associate in a much 
wider range of situations than the test situation. We expect that 
ability in the "Objects Seen" test is indicative of the capacity to 
perceive quickly and accurately a large number of perceptual fac- 
tors any one of which may later become significant. At any rate 
we entertain some such hopes as this unless we are willing, stu- 
pidly, to define "associative power" as ability in the "opposites 
test," and "quickness and accuracy of perception" as ability in 
the "objects Seen" test. But how are we to know whether our 
postulates are justifiable, our hopes fulfilled? Where is "Speed 
of Association," "Quickness of Perception" to be found? 

The answer, it would seem, is correlation. We must correlate 
performance in tests with performance in some larger line of 
activity which clearly is indicative of ability of one kind or 
another. It might perhaps be conceivable that we should find 
activities, suitable for this purpose, which are more or less clearly 
symptomatic of the various categories of the psychologist or 
even of the mental tester. But be that as it may, practically it 
is quite impossible. We are face to face here with another of 
the difficulties to which Whipple calls attention, the infinite com- 
plexity and variety of life. We cannot deal with functon di- 
rectly. We are obliged to deal with concrete specific fact. Fur- 
thermore the need of a mass of quantitative data confines us to 
the fields in which men engage 'en masse' and, preferably, where 
quantitative data are available. Practically, the science of Men- 
tal Tests is in its infancy in the field of education, and is in em- 
bryo in psychiatry, criminology, and industry. 

The data for this study were gathered in the field of education 
and we may proceed at once to the consideration of the criterion 
with which we are to correlate. We consider only two criteria. 

1 The term "function" is used loosely to indicate almost any kind of psycho- 
physical process. 



THE ANALYSIS OF MENTAL FUNCTIONS 3 

They are academic marks, and judgments of "intelligence." The 
latter are usually given by the same instructors who give the 
marks and occasionally by other judges alleged to be competent. 
Marks are a perfectly straightforward, objective form of achieve- 
ment and, in spite of the many objections which can be made to 
them, they have been used extensively on account of their avail- 
ability and on account of the quantitative form in which they are 
immediately given. Judgments have been used chiefly by those in- 
vestigators who are dissatisfied with marks as a measure of in- 
telligence, and they are supposed to reach "intelligence" in more 
direct if less objective fashion. Even the objection of lack of 
objectivity tends to disappear when the scale according to which 
judgments are made is the one originated by Karl Pearson to 
which Mr. Ruml has called the attention of psychologists in most 
interesting fashion. 2 However the choice of a criterion depends 
on the duty we expect that criterion perform, and inasmuch as 
my ideas on this subject are very different from those of Mr. 
Ruml, even though I also am inclined to prefer judgments to 
marks, it will pay us to look into that phase of the matter. 

Mr. Ruml argues that the criterion is, so to speak, an absolute 
criterion. If it does not measure "intelligence," it at any rate de- 
fines it, and thus becomes the sole measure of the value of the tests. 
If we do not take this position, he would urge, we are reasoning 
in a circle. If we decide on a given criterion because it correlates 
more highly with a given set of tests than some other criterion, 
we are choosing the criterion on the basis of the tests rather than 
the reverse. Now such considerations are perfectly sound if 
one is interested in measuring intelligence. The present writer is 
not. He prefers to analyze it, and he conceives the analysis of 
the factors which enter into some objective performance such as 
the obtaining of marks or judgments from an instructor to be 
so useful and interesting a prolegomenon that perhaps it may 
render the more ambitious task superfluous. For this reason he 
prefers the criterion which gives the highest correlation with a 
given set of tests, for high correlations give the analyst some- 
thing to work with, whereas, as we shall see, it is next to impos- 

2 Psychological Monograph, No. 105. 



4 CURT ROSENOW 

sible to reason from low correlations. The writer believes that 
judgments will be productive of higher correlations with "good" 
tests (note the circle) because he thinks that in practice judg- 
ments amount to little but a revision of the marks and the cor- 
rection of a few very flagrant cases, where achievement most 
evidently differs from ability to achieve. However this is a 
matter for future research to decide. Whatever criterion gives 
the highest correlation with tests will be best for purposes of 
analysis quite apart from a priori considerations. In the present 
study marks were used as a criterion because judgments were not 
available. Otherwise a comparison would have been made. 

If now we review the ground we have been over, it would 
seem that we are in a difficult position. On the one hand we 
have urged that tests should not be credited with symptomatic 
value for a priori reasons. On the other hand we have alleged 
that it is practically impossible to find other facts which do pos- 
sess such value. How then, we may be asked, can one set of facts 
whose meaning is unknown to us serve as a measure for another 
set about the significance of which we are equally ignorant? If 
the conditions have been represented correctly, the problem is 
indeed difficult. And yet it is precisely this problem with which 
the present paper deals. So it will be worth while to dwell a lit- 
tle on the question whether the problem does indeed take this 
form, even though, to many minds, the position will scarcely 
need defence. 

To anyone at all acquainted with the canons of scientific evi- 
dence it will be obvious that the good intention of the psycholo- 
gist who devises a "test" cannot be accepted in lieu of its symp- 
tomatic value. We may grant that the psychologist has knowl- 
edge of the workings of the human mind which, in some respects, 
goes beyond that of the man of affairs. We cannot grant that 
this knowledge, basic and fundamental though it may be, makes 
him a competent judge of specific social and practical efficiency 
or of general intelligence. To be sure, perception, memory, 
imagination, association, attention, generalization, etc., enter for- 
mally into every operation of the mind. But the relation of con- 
tent to form is one of the major problems of education and of 



THE ANALYSIS OF MENTAL FUNCTIONS 5 

psychology, not one of its fundamental laws whose formula we 
know and can apply with confidence. The whole problem of 
transfer of training and of formal discipline confronts us here. 
Some transfer, or better some identity of function in divers ac- 
tivities, is indeed a presupposition which is necessary so that tests 
may be thinkable. But to ascertain the kind and degree of iden- 
tity constitutes a problem which can be solved only by investiga- 
tion. 

Nor are things otherwise when we come to the consideration 
of the criterion. Whence can the authority be derived by virtue 
of which we credit any set of facts whatever with being a meas- 
ure of "general intelligence," "mental ability," or any other of 
the loose, vague terms which are in current usage ? How is the 
authority given, which enables us to take such a set of facts as a 
measure of specific ability? A moment's reflection will show 
that such authority is necessarily social. Now I do not think 
that a clear case can be made out for any of the criteria of intel- 
ligence which have been offered. The ordinary man is more 
likely, I think, to look upon excellence at school as the sign of 
special ability, rather than of general intelligence. And I ques- 
tion whether the average teacher working under average condi- 
tions is a competent judge of anything save average ability. 

On the other hand, academic marks are surely a measure of 
ability of some kind, and it is equally certain that it would be 
better to say — abilities. In other words the obtaining of marks is 
a complex achievement. Unless we accept the opinion of the 
"man on the street" or of the average teacher, we must analyze 
this achievement into its causal factors, if we wish to gain a bet- 
ter understanding of what it does stand for. Now such an analy- 
sis can be made in terms of tests. That is, the functions which 
are active in obtaining marks can be expressed in terms of the 
functions which are active in making scores in tests. And after 
such an analysis has been made, we shall be able to approach the 
problem of interpreting both the test and the criterion to better 
advantage. 28 In a later portion of this paper the results of such 

2a The advantage gained by such an analysis may not be obvious to some 
readers. I trust it will become so as we proceed. At present I will say that 



6 CURT ROSENOW 

an analysis will be presented with the purpose of showing what 
it really is and how it enables us to approach nearer to the prob- 
lem of determining the meaning of tests jointly with the analy- 
sis of the criterion. At present we must give our attention to 
the method of analysis. I may say at once that it is the method 
of partial correlation and that it is largely for the purpose of 
calling attention to its possibilities that the present paper is 
written. But as this method is relatively unfamiliar, and inas- 
much as my chief purpose is the arousal of interest, I have 
thought it best to keep the methodological discussion informal 
and non-technical. 

The position of the writer, then, up to this point, is that the 
end we should have in view in correlating tests with a criterion 
is the analysis both of the test and of the criterion. He protests 
against the naif assumption of the mental tester that the test is, 
ipso facto, of a definite, more or less simple mental function. 
And he protests even more strongly against the more sophisti- 
cated, and hence more dangerous contention of the psychologist 
that the criterion, even though it is complex and vague in its' 
significance, should be given artificial precision and simplicty 
by definition. The need is analysis. But it is a great deal easier 
to make this demand than to satisfy it. The difficulties with 
which the analyst has to deal lead us to considerations of another 
kind. 

II. Discussion of the Method 

The first difficulty which we encounter is wholly artificial. The 
statement is often made by writers with a tender metaphysical 
conscience that a coefficient of correlation tells us nothing of true 
causal relations. Be that as it may, it does tell us quite as much 
about them as any other quantitative statement aiming at rep- 
resenting relations. For example, v = gt, such writers would 
say, tells us nothing of the nature of gravity. It informs us 
merely that, on this earth, the velocity of falling bodies varies 

I conceive the process of acquiring knowledge of the symptomatic worth of 
tests as a growth. If we desire logical demonstration, we must put into our 
definitions what we mean to take out of them. 



THE ANALYSIS OF MENTAL FUNCTIONS 7 

directly with the time, and that 'g' is the increment of velocity 
corresponding to an increment of time. Now, as we shall see, 
v = gt is nothing but a regression equation. We can substitute 
for V performance at the criterion, for ( t l performance at the 
test, and for 'g the coefficient of regression, and we are furnished 
with information which is analogous, though not perfectly so, to 
the case of falling bodies. One difference is that in the one case 
further interpretation of the "law" is centuries old and familiar, 
and it therefore seems obvious and simple. In the other case we 
do not, in most cases, have any very plausible interpretation 
which goes beyond the facts. 

We encounter a more serious difficulty in the following. The 
statement is frequently made that the coefficient of correlation is 
meaningless in the case of non-linear regression. 3 If this be 
true, the usefulness of correlation as a means of psychological 
analysis is seriously curtailed, for cases of strictly linear regres- 
sion are rare. The "proof" of linearity simply means that non- 
linearity cannot be proven. 4 It does not and cannot show that 
a straight line is the most probable regression, unless the line, 
(or lines), which passes through the means of the arrays is (are) 
actually a straight line. (The reader who is unfamiliar with this 
terminology is asked to reread this passage after he has read the 
next few pages.) Linearity is then assumed on account of its 
practical workability. But, fortunately, even when non-linearity 
can be proven, the statement is not true that the coefficient of 
correlation becomes meaningless. It merely loses some of its 
meaning. It is one of the great contributions of Yule to have 
shown the precise significance of the coefficient of correlation 
under any and all circumstances. This leads us to the consider- 
ation of points of a more technical kind. 

In what follows it is assumed that the reader has a slight de- 
gree of familiarity with the terminology and the mathematical 
theory of correlation. Although the discussion is elementary, 
it does not aim to be an elementary exposition of the subject. 

3 Brown. Essentials of Mental Measurement. Pp. 44-45. 

4 The test for linearity is a farce with say one hundred observations, for 
the value of rj can be made to vary within wide limits, for a single set of 
data, by varying the magnitude of the class-interval. 



8 CURT ROSENOW 

Still less does it pretend to give mathematical proofs. The 
reader who wishes to go into that phase of the subject is referred 
to the literature, and specific references will be made in their 
proper place. The aim is merely to present as simply as may be 
the points which the writer conceives to be essential for his 
purpose. Indeed if it should turn out that the reader who has 
had no acquaintance at all with the subject can follow the argu- 
ment, the writer will be gratified. On the other hand I wish to 
guard very carefully against creating the impression that I am 
trying to pose as an expert mathematician. I conceive myself to 
have barely enough proficiency so that I can seize on some of the 
essentials and attempt some manipulation of a simpler sort. I 
am particularly anxious to make this acknowledgment in view 
of a number of instances where it should be made and is not. 

Let us now suppose that a number of observed concomitant 
variations of two variables are expressed as deviations from 
their respective means and are then plotted, using rectangular 
coordinates, the means of the variables being at the origin of 
coordinates. Let us suppose, furthermore, that the points so 
plotted form a smooth continuous curve of some kind. This is 
a state of affairs approximated in the exact sciences. If now we 
are able to determine the equation of this curve, it is clear that 
we have in such an equation an expression which portrays ac- 
curately the amount of concomitant variation which actually oc- 
curs. For example, if we were to plot the results of a number 
of observations on the distance which a freely falling body covers 
during varying durations of time, we would find that the points 
would form a parabola, whose equation would enable us to esti- 
mate distance from time and vice versa. Suppose now that in- 
stead of a smooth curve our points form a jagged irregular line, 
and that it is required to find the straight line which most closely 
approximates the actual line. This may be done by the method 
of least squares, i.e., upon the condition that the sum of the 
squares of the deviations from the straight line be a minimum. 
It may be shown that the line which satisfies this condition passes 
through the origin of coordinates. Its equation therefore will 
be x = by, where b is the tangent of the angle which the straight 



THE ANALYSIS OF MENTAL FUNCTIONS 9 

line makes with the y axis. This equation may then serve ap- 
proximately the same purpose as the equation for the parabola in 
the case of falling bodies. It represents, with absolute accuracy, 
the average amount of concomitant variation exhibited by our 
data. And if its use yields us a set of individual values which 
approximate the truth to the point of useful approximation, "re- 
gression" is said to be rectilinear. (Needless to say, this is not 
intended to be a technical definition of regression. I believe how- 
ever that it gives a faithful account of its meaning). 

Any curve or line which "fits" the data to the point of useful 
approximation is called a curve of regression. Such a line satis- 
fies the condition of least squares for some given type (shape) 
of line. It is often said to be the line which passes through the 
means of the columns or rows respectively. Such a line ob- 
viously satisfies the condition of least squares. As has just been 
said, if it is straight, regression is said to be recti-linear or, 
briefly, linear. Its equation is the regression equation. Its tan- 
gent with the appropriate axis is the coefficient of regression. 
There are, of course, two such lines, one passing through the 
means of the x arrays, the other through the means of the y 
arrays. 

We are not at all concerned, for our present purposes, with 
ways and means of evaluating the coefficient of regression. It 
is important to point out that the coefficient of correlation and 
the coefficient of regression are identical in value, when the devi- 
ation of the variables from their means are expressed in terms 
of their respective standard deviations as the unit of measure- 
ment. It follows that the meaning of the coefficient of correla- 
tion, in so far as it is a means of diagnosis and a measure of re- 
lation, is exhausted by the meaning of the coefficient of regres- 
sion. Indeed it is possible to look on the coefficient of correlation 
as a convenient algebraic expression which enables us to find the 
value of the regression coefficient, to establish its validity or the 
degree of confidence which may be given to it, and to show the 
magnitude of error which may be looked for when it is used for 
prediction and diagnosis. 

With this in mind let us now enter upon the closer examina- 



io CURT ROSEN 0W 

tion of the claim that the coefficient of correlation is meaningless 
in cases of non-linear regression. In the first place we may ad- 
mit without argument that the coefficient is meaningless with 
respect to the form of the true relation existing between the vari- 
ables. If it is desired to describe the form, there can be no pos- 
sible meaning in describing say a parabola by means of a straight 
line. Next we may note, without formal proof, that the coeffi- 
cient will of necessity have a lower value than an expression 
which measures the amount of deviation from a curve of closer 
fit in analogous terms. 5 It follows that, taken merely as an indi- 
cation that an actual relation does exist between two variables, 
r, the coefficient of correlation, is actually entitled to increased 
confidence if non-linear regression is shown. Indeed the mere 
proof of non-linear regression is in and of itself proof of the 
existence of a true relation, and also of the fact that it is greater 
than indicated by r. It can hardly be claimed that a positive as- 
sertion which errs only on the conservative side is meaningless. 
As a special case we may note that r = o does not necessarily 
indicate the absence of relation. 

Again, in the case of strictly linear regression, the errors of 
estimate will be equal for every part of the line and will be sym- 
metrical as to sign. 6 In the case of non-linear regression this 
will not be the case. For example, if the "true" regression is a 
sine-curve, the errors of estimate will be least at the points of 
intersection with the straight line of best fit, and will tend to be 
of opposite sign at different parts of the line. In short, the 
errors will tend to be systematic. But, when all is said and done, 
the straight line of best fit is what it claims to be, and, in the case 
of more than two variables, predictions and analyses made by its 
use will, on the average, be closer to the truth than any other 
conclusion based on the data and arrived at by any of the prac- 
tically possible means. The subject of non-linear regression for 
the psychologist amounts simply to this. If he is investigating 
the relation of two variables to each other he can get nearest to 

5 Such an expression is the correlation ratio, rj- It does not, however, 
give us the equation of the curve of closer fit and hence is of no use for 
diagnosis. 

6 The statement is true only for the types of regression usually found. 



THE ANALYSIS OF MENTAL FUNCTIONS n 

the truth by "fitting" a curve and determining its equation. Even 
in that case useful results are practically always obtainable by 
assuming linearity. But if one is dealing with a complex situa- 
tion the only practical possibility with our present technique is 
to assume linearity. The results, when properly interpreted, will 
not be meaningless. 

Suppose now, to take a hypothetical example, that we wish to 
ascertain the relation between crop yield and water supply in 
connection with a proposed irrigation scheme. We have avail- 
able for the purpose data on the concomitant variation of rain- 
fall and crop yield, and we find the coefficient of correlation of 
these two variables to be 0.40. 7 We are then able to estimate the 
most probable crop yield we may expect from supplying a certain 
amount of water per acre. We will also be able to estimate, to 
any desired degree of probability, within what limits the actual 
yield will fall. In the case given, for any respectable degree of 
probability, the limits will be so wide as to render the informa- 
tion of little practical use. Now it may be argued that the re- 
lation between water supply and crop yield in a climate in which 
the sun is always shining is different from the relation which 
exists where water supply and sunshine very probably are in- 
versely related. We wish to know the relation between yield 
and moisture supplied when there is a constant amount of sun- 
shine, for this relation may be very much closer. If now in ad- 
dition to our other data, we have data on the concomitant varia- 
bility of "sunshine," we shall be able to supply this information. 
For suppose that we use the regression equation describing the 
relation between sunshine and yield to estimate the yield per acre. 
We estimate say 60 bushels of wheat. Actually it turns out to be 
30 bushels. The difference then is an error of estimate. But, 
as we shall see at once, it is more than that. For consider that 
the correlation which we found, however low or high its prob- 
ability, is in any case more probable than any other value known 
to us at present. Were it the "true" value, the error of estimate 
would be due exclusively to causes other than variation in the 

7 The illustration, greatly modified and expanded, is taken from Yule's 
Introduction to the Theory of Statistics. 



12 CURT ROSENOW 

amount of sunshine. It would, in fact, be an exact measure of 
the total effect of all operative causes, with the effect of sunshine 
eliminated. As it is, it is just such a measure to the highest de- 
gree of probability possible to us with our present technique, and 
on the basis of the data at hand. We may therefore call an error 
of estimate a residual. Similarly, if we estimate the amount of 
rainfall during a given period from the amount of sunshine dur- 
ing that period, the residual will be a measure of the amount of 
rainfall associated during the period with all facts other than 
the somewhat obvious one that the sun is not shining when ob- 
structed by clouds. (Obvious as it is, the fact is not irrelevant. 
At any rate, the reader is asked to fix his attention on the fact 
that a residual is a measure or representation of the association 
which exists between the fact we wish to estimate, and all other 
associated facts except one.) If now we actually compute all of 
the residuals which represent the relation of yield to all facts 
other than sunshine, and also all of the residuals representing the 
relation of rainfall to all facts other than sunshine, we shall have 
two sets of measures strictly analogous to our original data on 
the concomitant variation of yield and rainfall, except that the 
effect of sunshine has been eliminated from both measures. Now, 
if we compute the coefficient of correlation from these data, we 
shall have a measure of the relation which exists between yield 
and rainfall, the effect of sunshine being eliminated. In the no- 
tation of Yule, 8 if crop yield is X l5 rainfall X 2 , and sunshine X 3 , 
we shall have the value of r 12 . 3 . 

8 It may be desirable to give some account of this notation, sufficient to 
render its use in the text intelligible. All variables are denoted by sub- 
scripts of X, such as Xi, X 2 , etc. The coefficient of correlation between any 
two is indicated by writing their subscripts beneath the symbol r, e.g. r 12 . 
The coefficient of correlation between two variables, after the influence of 
other variables has been eliminated, is called a coefficient of partial correla- 
tion, and is written as follows. Let the variables whose relation is being 
expressed be X t and X 2 , and let the variables which have been eliminated be 

X 3 , X 4 , X 5 Xn. Then the coefficient of partial correlation is written 

1*12.345 n. The subscripts denoting the variables whose relation is being 

expressed are called "primary" subscripts and are written to the left of the 
point. Those denoting the eliminated variables are called "secondary" and 
are written to the right, r^ is called a coefficient of zero order, r 12 . 3 is a 
coefficient of the first order, r^ . 34 of the second order, etc. In general, the 



THE ANALYSIS OF MENTAL FUNCTIONS 13 

To be sure, this is not the method which is actually used. If 
it were, the amount of arithmetic would be almost infinite in 
complex cases. But Yule has shown that the method which is 
used is equivalent to such a method both in meaning and result. 9 
And, to my mind, this fact shows in the clearest fashion the 
meaning of partial correlation. Indeed it is largely for the sake 
of this single point which shows, I think, the simple causal reas- 
oning, the simple logic, which underlies the complex mathematics 
which befuddles us, that the previous discussion has been given. 

It follows that every claim we have made for the coefficient of 
correlation when only two variables are taken into account, is 
valid for the partial coefficient. That is, (1) Its meaning in 
cases of non-linear regression is clear and definite, (2) Its val- 
idity or "significance can be computed in the ordinary way, (3) 
The probable magnitude of average error incident to its use can 
be computed in the ordinary way. Indeed, as Brown says and 
says truly, "The full significance of correlation in psychology is 
to be found in the general theory of multiple correlation, of 
which the correlation of two variables is only a special case." 10 

We may express the relation of crop yield to both rainfall and 
sunshine in a single equation by simply adding the yield due to 
rain with sunshine constant to the yield due to sunshine with rain 
constant. Obviously such a sum is due to the combined effect of 
the two. That is, x x = b 12 . 3 x 2 + b 13 . 2 x 3 . This is a partial 
regression equation. It would be represented graphically by a 
plane. Now the differences between the values of x x obtained 

order of a coefficient denotes the number of secondary subscripts. The 
coefficient of multiple correlation, the meaning of which will be discussed 
below, denotes the relation which exists between a single variable, and the 
results which are obtained by estimating the values of that variable from a 
number of others by means of the regression equation. Its symbol is R, and 
is not to be confused with Spearman's R. The single variable is called 
"dependent," the others "independent." R is written Ri( 2 s4.- .») where 1 
is the subscript of the dependent variable, 2, 3, 4, etc., the subscripts of the 
independent variables. 

9 G. U. Yule. Proc. Roy. Soc. Series A, vol. 79, 1907. "On the theory of 
normal correlation for any number of variables treated by a new system of 
notation. 

10 Wm. Brown. Essentials of Mental Measurement, p. 128. 



14 CURT ROSEN OW 

by the use of this equation, and the actual values of xi, will be 
residuals containing the part of x x not associated with either 
x 2 or x 3 . These residuals might in turn be used to find the rela- 
tion of a fourth variable to Xi, with x 2 and x 3 eliminated, and 
so on indefinitely. 

As has been said, the correlation which exists between the 
actual values of Xi and the values estimated from an equation 
of partial regression is called multiple correlation, and the sym- 
bol of its coefficient is R. It is a measure of the closeness with 
which Xi can be estimated from x 2 , x 3 , etc. It has some very 
useful properties which are important for the purposes of this 
paper and which we will discuss later. 

Before leaving this part of the discussion, I feel that I am 
under moral obligation to call the attention of the reader to a 
statement by the highest authority on the theory of correlation, 
Prof. Karl Pearson. "The method (multiple correlation) . . . 
does assume that linearity applies within the degree of useful 
approximation. . . . The general linearity ought to be tested in 
all cases. Nothing can be learned of association by assuming 
linearity in a case with a regression line like A, much in a 
case like B." 11 (See diagram, page 15.) I wish to say 
that I realize fully my audacity — perhaps impertinence would be 
a more fitting phrase — in commenting on this statement. I am 
perfectly willing to follow blindly any course indicated by Karl 
Pearson. But, I am equally willing to do this when Mr. Yule 
leads the way. Now it does not seem to me that there is any 
real conflict of authority here and my interest in the subject 
compels me to point this out. It seems to me that the issue 
hinges on the meaning of the phrase "point of useful approxima- 
tion." Mr. Yule has shown, and so far as I know his proof has 
not been challenged, that r retains an average significance under 
any and all conditions having to do with regression. If this be 
useful, the assumption of linearity is legitimate provided only an 
average significance is attached to the result. In a case like A 
there isn't any linear association. But if now the average slope 

11 K. Pearson. Biom. vol. 8, 1911-1912, p. 439. "On the general theory of 
the influence of selection on correlation and variation." 



THE ANALYSIS OF MENTAL FUNCTIONS 



15 



I 




e 



c 



of the curve be changed (see diagram C) there will be such 
association and r will be its measure. The absence of associa- 
tion, on the average, was due to its real absence, not to the form 
of the regression. Whether or no such results are "useful" will 
be determined by the special conditions of the particular problem 
in hand. At any rate, they have a definite meaning. 

Before leaving the topic, let me repeat what I said at its intro- 
duction. I have tried throughout to make clear the meaning of 
certain phases of the topic which I deem essential for my pur- 
poses. I have not tried to prove anything. The reader has been 
asked to accept the statements made on the authority of Yule. 12 

What now is the significance of partial correlation for the 
Mental Test situation? In the first place it furnishes us with 
the only means at present available for the analysis of the test 
and the criterion. For suppose that we find a correlation of 
+0.28 for the Logical Memory test with academic marks. The 
previous discussion should have made it clear that this is not a 
measure of the extent to which "logical memory" is a factor in 
"general intelligence." As any instructor will testify, marks 
are not even a very good measure of academic intelligence, and a 



12 In addition to references given above, See G. U. Yule. Proc. Roy. Soc. 
Vol. 60, 1897. "On the significance of Bravais' formulae for regression, 
etc. - - ." 



16 CURT ROSEN OW 

cursory examination of the test in say Whipple's Manual will 
show that most of the logic is contained in the name. This is 
further emphasized when the test is evaluated quantitatively, as 
it must be for purposes of correlation. Mechanical memory is 
by no means confined to the learning of nonsense syllables and 
digits. If it were, it would not be a problem for education. I 
can conceive of at least the possibility of so organizing a non- 
sense syllable test that, as a criterion of logical memory, it would 
be superior to the test which bears that name. At any rate, it is 
not all certain, a priori, that a student who depends largely on a 
rather verbal and mechanical type of memory is at a serious dis- 
advantage either in this test or in the matter of marks. If now 
we are able to devise a test which, a priori, carries a somewhat 
stronger presumption of logical memory, we have at any rate 
some material for analysis. For let X x stand for Marks, X 2 for 
the original logical memory test, and X 3 for the supposedly more 
logical test, then r 12 . 3 carries a stronger imputation of being 
rather verbal than r 12 , and there is a greater probability that 
r 13 . 2 stands for a more logical type. For that which the two 
have in common has been eliminated in each case in so far as it 
is associated with marks, and therefore the relation to marks 
of that which is peculiar to it stands out more clearly. We see 
that we have here the promise of a fruitful way of combining 
experimental and statistical research. For a slight modification 
of the test, or even a change in the method of scoring, may lead 
to results of significance with regard to the nature of the test. 
This will be illustrated later. Speaking generally, a partial coeffi- 
cient (r 12 . 3 4 . . • n ) is more easily interpreted than a coefficient 
of zero order (r 12 ), for in the case of r 12 we have to face the 
vague question why X 2 is related to X 2 , while in the other case 
we may ask what in X 2 is related to X x that is peculiar to it, and 
has nothing to do with X 3 , X 4 , etc. In other words, we have 
more data on which to base analysis. 

Another use to which partial correlation might be put is in 
connection with so-called "practical" diagnostic work. If a num- 
ber of tests have been given, it is usually desired to combine them 
into a single measure of their diagnostic value, with reference 



THE ANALYSIS OF MENTAL FUNCTIONS 17 

to a single criterion such as marks. The method sometimes 
used is that of expressing all the scores as deviations from their 
mean with their respective standard deviation as the unit of 
measurement. The scores for each test for each individual are 
then added, and these combination scores are correlated with the 
criterion. 13 The method is somewhat faulty. It attaches equal 
importance to all tests regardless of their correlation with say 
marks, and it is perfectly obvious that this is false. Worse, when 
handled wrongly, it may even serve to conceal linear relations 
easily discernible in the data. For, to take an extreme case, sup- 
pose we combine r 12 = + 1.00 and r i3 = — 1.00 according to 
this method. The result, if the two distributions happen to be 
nearly parallel, will approximate zero, and we shall have suc- 
ceeded in converting two perfect diagnostic tools into an abso- 
lutely useless one. This source of error can be obviated by com- 
puting all correlations of type r and reversing the signs of the 
scores of all tests showing a negative correlation, reversing the 
meaning to correspond. At its best, the method is purely em- 
pirical and the meaning of the coefficient obtained by its use is 
neither clear nor definite (except mathematically). We cannot 
reason from it, we cannot use it as an analytic tool. If it does 
not possess diagnostic value, it possesses no value whatever. 

The best method of combining a number of tests is to find 
their regression equation. R, the coefficient of multiple corre- 
lation, will be the indirect measure of the diagnostic value of 
this equation. As we shall see, R is exceedingly valuable for 
analysis, is more easily calculated than the regression equation, 
and, unless we wish to apply diagnosis to the case of single indi- 
viduals, it renders the finding of the regression equation unneces- 
sary. In so far as he knows, the use of R for such purposes is 
original with the writer. When regression is truly linear, R 
gives us, within the limits of accuracy of sampling, a measure of 
the actual relation which exists between two complex set of facts, 
the criterion and the tests. Besides it will enable us to make an 
accurate analysis of the criterion in terms of the tests, in so far 
as it is associated with the tests. When regression is non-linear, 

13 R. S. Woodworth. Psy. Rev., 1912, p. 97. 



18 CURT ROSENOW 

the results have the same significance to a lesser degree of ap- 
proximation. 

In view of all this, it may seem somewhat surprising that the 
method has not found more frequent application. One reason 
for this is perhaps that the subject is somewhat difficult and not 
well understood generally. Another reason is probably the large 
amount of arithmetical labor called for in complex cases. Yule 
states that the working out of a case involving eight variables is 
practically beyond the powers of a single individual. 14 Kelley 15 
states that in the case of eight variables it is practically necessary 
to resort to an approximation. 16 But Kelley himself has reduced 
the amount of mechanical labor materially through the publica- 
tion of a very useful set of tables, contained in the bulletin re- 
ferred to. I myself have devised a scheme of procedure, involv- 
ing the use of R, which makes it possible to reach all of the re- 
sults which Kelley reaches, and perhaps a little more, with ap- 
proximately half the work indicated by him. A full exposition 
of this schema will be found in the appendix. These mechanical 
improvements, as well as the fact that a complete working out 
of all possible relations is not necessary, bring the method within 
the reach of a single individual. For the purposes of the present 
paper I worked out a case of sixteen variables without resorting 
to approximation. It took me a little over two months, but I did 
a lot of useless work. I could do this work now in five or six 
weeks at the most. Of course I was working with a compara- 
tively small number of observations, but, after the coefficients of 
zero order have been found, the amount of arithmetic does not 
depend on the number of observations. But be that as it may, 
the method must come into use if scientific analysis is ever to take 
the place of blind fumbling about. For it is the only method 

14 G. U. Yule. Roy. Stat. Soc. Journ., vol. 60, p. 182. 

15 Kelley is the only American psychologist who has exploited the method 
of partial correlation. See his "Educational Guidance," Teachers College, 
Columbia University, Contributions to Education, No. 71. In the opinion of 
the writer, Kelley's work loses some of its value through his failure to call at- 
tention to some of the difficulties and sources of misunderstanding with 
which the subject is hedged about. These difficulties are still in front of us. 

16 Bull. No. 27, U. of Tex., May 1916, p. 18. 



THE ANALYSIS OF MENTAL FUNCTIONS 19 

available at present that holds out even a hope of making syste- 
matic progress in attacking situations as complex as those with 
which we have to deal. 

For the present however let us dwell rather on the limitations 
of the method. In the first place it would be quite erroneous to 
suppose that the magnitude of R, and consequently its value, can 
be indefinitely increased by simply increasing the number of 
tests. To take another illustration from Yule, 17 if r 12 = 0.8, 
r 13 = 0.4, and r 23 = 0.5, it would be quite natural to suppose 
that Xi could be estimated with greater accuracy from both X 2 
and X 3 , than from X 2 alone. But this would be quite wrong, be- 
cause 1*13 . 2 = o. In other words, everything in X 3 that has 
diagnostic value, or that is associated with X lf is contained in 
X 2 . Pearson dwells at length on this point. 18 For example, if 
an infinite number of variables are correlated equally with each 
other, the value of r being 0.5, he shows that R with reference 
to any one of them is 0.71. In the case of ten such variables, 
R = 0.67, for five variables, R = 0.65. The difference in diag- 
nostic value between 0.65 and 0.71 is negligible. How serious 
a difficulty this is will appear later. At present let us note that the 
trouble is not with the method, but with the material with which 
it deals. 

The other difficulty to which I wish to call attention is one of 
interpretation. R is essentially positive regardless of the signs 
of the coefficients of the regression equation. It is therefore 
subject to biased error, that is, errors due to fluctuations of 
sampling will not tend to neutralize each other, but will be cumu- 
lative. Consequently the "probable error" of R is not a true 
measure of its validity. It should be compared to the value of 
R in the case of a number of really uncorrelated variables owing 
to fluctuations of sampling alone. Pearson, in the article re- 
ferred to below, promises us a formula for finding such a value, 
but, to date, I have not found it in the literature accessible to 

17 Intro, to the Theory of Statistics, p. 237. 

18 K. Pearson. Biom. vol. 10, 1914-15, p. 181. "On certain errors with re- 
gard to Multiple Correlation occasionally made by those who have not ade- 
quately studied the subject." 



20 CURT ROSENOW 

me. 19 Yule 20 publishes an approximation formula which gives the 
value. It is (n — i )** / N**, where n is the number of variables, 
and N the number of observations. Inspection of this formula 
will show that its value may easily become unpleasantly large. 
For example, if n = 16, and N = 92, as is the case in the prob- 
lem I worked out in the present paper, then R === 0.40 ± 0.06. 
That is, although R is seven times as large as its probable error, 
it has no validity whatsoever. It is the failure to call attention 
to this fact which I alluded to in discussing the work of Kelley. 

III. Application of the Method to Correct Data 

In view of these two serious limitations I might perhaps be 
asked why I undertook so laborious a task as the computation 
which forms a part of this paper. The answer is simply that it 
is no part of my purpose to advertise a method for making a silk 
purse out of unsuitable material. There is in this paper no at- 
tempt to break the record for altitude. The value of R does not 
interest me except in so far as it is an instrument of analysis. 
My object was to sift a mass of typical material down to its sig- 
nificant constituents. The general character of the results I was 
fairly sure of before I undertook the work, but I could not know 
precisely what material would be retained by the meshes of the 
sieve. I could have reduced the number of variables to eight or 
nine and have been morally certain that I was not discarding any- 
thing of value, but it is questionable what weight my moral cer- 
tainty would have carried with others. Besides I fancied that a 
drastic concrete illustration of the difficulties which I have just 
called attention to would do no harm. Furthermore I am very 
much interested in the subject of partial correlation and hope 
that the present paper, in conjunction with the appendix, will 
have a methodological value. The particular material was used 
simply because, owing to the kindness and scientific attitude of 
Dr. Kitson, it was available. But tests are not the only things 
capable of analysis. Nor are they the only source of information 
on which diagnosis can be based. For example, if Kelley 's work 

19 Pearson, Biom. vol. 8, p. 437, op. cit., p. 18. 

20 Proc. Roy. Soc. 1907. Op. cit. 



THE ANALYSIS OF MENTAL FUNCTIONS 21 

were based on more than thirty-three observations, it would show 
that the six tests which he uses and combines with academic 
marks in grammar school and with judgments of ability by the 
teachers — are negligible for the purpose of predicting perform- 
ance in High School in comparison with these other means of 
diagnosis. 21 However I ought to say that up to a time when the 
work was more than half finished I thought I would have a con- 
siderably greater number of observations at my disposal than 
proved to be the case. This brings us to the consideration of the 
material with which the present investigation deals. 

This material was put at my disposition by Dr. Kitson. 
Neither he nor I dared to hope that the investigation would lead 
to results of final validity. In the first place the number of ob- 
servations were not adequate. Besides Dr. Ktson himself looks 
upon the stage of his work from which the data were taken as 
its pioneer stage. Since that time he has added new tests and 
has improved the old tests. If it had been possible to include 
this later material, there is every reason to believe that more 
significant results would have been obtained. I do believe that 
needless duplication of identical functions is a feature of all 
lists of tests in actual use if they are at all extensive. In order 
to enable me to test this out, Dr. Kitson furnished me with such 
material as he had available. Even though the results do not 
have a direct bearing on his later work, or on similar work done 
by others, it was thought that the indirect light they would cast 
would have value. 

The material then is the same as that obtained by Dr. Kitson 
for his "Scientific Study of the College Student." 22 As may be 
seen by referring to this monog-raph, Dr. Kitson gave a large 
number of tests to the students of the college of Commerce and 
Administration at the University of Chicago. The work there 
described covers a period of two years. It has been continued 
and the results of two more years are now available. At the 
time my own work was done, the academic marks for the fourth 
year were not at hand. Besides so many changes had been made 

21 T. L. Kelley. Ed. Guid., p. 71 ff. 
22 Psy. Rev. Mono., 1917, vol. 23, No. 1. 



22 CURT ROSENOW 

in the tests themselves that their combination with the other 
three years did not seem feasible. There did not, however, ap- 
pear to be any a priori reason why the first three years could not 
be combined, so that I expected to have 150 sets of observations 
on which to base any conclusions I might reach. I did not in- 
vestigate the three years separately until, when the work was over 
half finished, certain differences forced themselves on my atten- 
tion. Then I did investigate and found, amongst other differ- 
ences, that the academic marks for the third year differed sig- 
nificantly from those of the other two years. Inquiry amongst 
the members of the faculty resulted in conflicting evidence as to 
the reasons for this, so that I was compelled, regretfully, to dis- 
card these data. This reduced the number of observations, or sub- 
jects, to 92, and the period covered became precisely the period 
described in Dr. Kitson's Monograph, i.e., the academic years 
1913-14, and 1914-15- 

In this group there are included 80 freshmen and 12 sopho- 
mores; 39 freshmen and 6 sophomores in 1913, and 41 freshmen 
and 6 sophomores in 191 4. The desirability of the inclusion of 
the sophomores may be questioned. However it is the practice 
at the School of Commerce and Administration to test all stu- 
dents who have not previously had the tests, and in such a group 
there always are a certain number of individuals who come from 
other departments, or institutions with advanced standing. Con- 
sequently such a group is representative in a definite sense. 

As the Psychological Review Monographs are quite accessible, 
I deem myself absolved from the uninteresting task of copying 
the full description of these tests. Most of them are standard 
and well known. Also their names give a good indication if 
their general nature. As much additional description as seems 
essential will be given informally with the discussion. At pres- 
ent I give only the names in conjunction with the numbers which 
they were given in the present study, and a brief description of 
the more important tests. The list follows. As will be seen, the 
criterion, academic marks, is given No. 1. 

(1) Academic Marks. 

(2) Immediate memory for logical material, heard. 



THE ANALYSIS OF MENTAL FUNCTIONS 23 

(3) Immediate memory for logical material, seen. 

(4) Loss or gain of log. mat, heard, after two weeks. 

(5) Loss or gain of log. mat., seen. 

(6) Sentences Built. 

(7) Hard Directions, printed, speed. 

(8) Constant Increment, speed. 

(9) Memory for objects seen. 

(10) Number-checking (cancellation). 

(11) Opposites, speed. 

(12) Memory for numbers heard, (span). 

(13) Word building. 

(14) Opposites, accuracy. 

(15) Constant Increment, accuracy. 

(16) Hard Directions, accuracy. 

The description of the seven tests which will be of most in- 
terest to us follows : 

No. 2. Logical Memory, immediate, auditory. 

Materials : Blank sheet of paper and pencil. 

Directions : "I am going to read you a rather long passage and shall ask 
you to listen very carefully, for when I have finished I wish you to 
reproduce the meaning of the passage. The passage is too long for 
you to remember word for word, but try to get the entire meaning, 
then in reproducing, use the same words as appear in the text when- 
ever you can. 

The Passage : The passage may be characterized as popular science. 

Method of Scoring: It will be noted that this passage contains a main 
proposition and three illustrations, the last one of which is amplified. 
For reproduction of the main proposition, two units were given; for 
mention of the first, second, and third illustration there were given 14, 
13, and 14 units respectively. Thus by merely stating the main propo- 
sition and the illustrations the individual could score 43. In addition 
to these gross divisions, the passage was further divided into 81 ideas. 
Counting each one of these as two thirds of a unit, their united value 
is 54, which added to the 43 unit mentioned, permits scoring on a 
basis of 97 points for correct reproduction of the passage." 

No. 4. Logical Memory, immediate, visual. 

Materials : See directions. 

Directions: "On the reverse side of the paper before you will be found 
a long passage which I wish you to read carefully when I give the 
signal. Read it but once, then turn it over, and on the back of it 
write all you can recall of the passage. Be careful to read each sen- 
tence but once, then turn over the paper and reproduce the meaning as 
accurately as possible." 

The Passage: May be characterized as popular psychology. 

Scoring: Same as in No. 2. 



24 CURT ROSENOW 

No. 3. Loss or gain, Logical Memory, auditory. 
Direction: Write all you can recall of the passage I read to you at the 
last psychological examination, beginning "More than once it has hap- 
pened in the history of science." 
Scoring: The papers were first scored as in two. Then the difference 
between No. 2 and No. 4 was taken as the score of No. 4. 

No. 5. Loss or gain, Logical Memory, visual. 

Analogous to No. 3 in every way. 
No. 9. Memory for Objects, visual. 
Materials : Covered box twelve by twenty-three inches, containing the 
following objects fastened to the bottom: fountain-pen, pencil, twenty- 
five cent piece, envelope, inkwell, maroon ribbon, ruler, pen-filler, two- 
cent stamp, and key. 
Directions: I am going to show you a group of objects for six seconds, 

then will ask you to name them aloud from memory. 
Scoring: The score represents the number of objects correctly reproduced. 

No. 6. Sentences Built. 

Directions : I will give you five minutes in which to make as many sen- 
tences as possible containing three words which I will give you pres- 
ently. For example, if I gave you the words money, river, Chicago, 
you might make a sentence like this: "Chicago spends much money 
improving its river." You may use either singular or plural forms of 
the words, nominative, objective, or possessive case. Simply use all 
three of the words in a sensible sentence and make as many different 
sentences as possible. The three words are, — citizen, horse, decree." 

Scoring : The score represents the number of sentences formed. 

Extract from Dr. Kitson's comment : "... the papers which contained a 
relatively large number of sentences necessarily showed much same- 
ness in subject matter and structure." 

No. 8. Constant Increment. 

Material : Card containing one hundred two-place numbers. 

Directions : I am going to give you a list of 100 numbers and shall ask you 
to add four to each number as quickly as possible, giving the sum 
aloud. You may practice on this list : 22, 34, 92. Begin at the top of 
each of the four columns and add four to each number. You need not 
be afraid to go fast for the test is easy and you are not likely to 
make mistakes. You should be accurate, however, because every error 
will take off one point from your score. The main thing is to add as 
rapidly as possible. 

Scoring: The number of errors was the accuracy score. The number of 
seconds, the time score. 

The materials accessible to me were the gross scores in all of 
the fifteen tests. In no case were the original records at hand, as 
they had been destroyed some time before. Had they been avail- 



THE ANALYSIS OF MENTAL FUNCTIONS 25 

able, it would have been possible to push analysis further than 
was actually the case, and many of the suggestions made below 
could have been investigated. I was not however obliged to ac- 
cept any empirical indices, as all indices which Dr. Kitson used 
were based on the gross scores. 

This, practically, is all the information we need for the present. 
Now our problem in evaluating these tests is not simply that of 
determining their total relation to marks. If it were, we could 
solve it easily and directly by simply computing the fifteen corre- 
lations which indicate this relation. We should find, for example, 
that the correlation of the Auditory Logical Memory test with 
marks is + 0.28, and that of Hard Directions with marks is 
+ 0.25, or using the numbers assigned to these tests, r 12 = + 
0.28, r 17 = + 0.25. But such numbers would tell us nothing of 
the nature of the relations in each case. The functions measured 
by r 12 and r 17 might in reality be identical, independent, or even 
mutually exclusive. There would be little sense in debating such 
an issue on a priori grounds when we can find directly that 
r 27 = + 0.30. We now have the beginnings of analysis, for we 
know, to the degree of probability of which our data permit, that 
the two functions are different in some respects and identical in 
others. A similar line of reasoning might be applied to every 
one of these fifteen tests, or variables, paired successively with 
each one of the others. In table 1 there will be found 
a complete list of all possible correlations of our fifteen tests 
amongst themselves and with marks, 120 all told. They are given 
in full because, aside from the raw data which are too bulky to 
print, they represent the complete data for this study. But, aside 
from this, the reader will be able to convince himself by a study of 
this table that the data in their present shape are far too complex 
to permit of conclusions more definite than the very vague one 
we have just stated with reference to No. 2 and No. 7, i.e., that 
they are alike in some respects and different in others. (Of 
course, if we have had practice with the method of partial cor- 
relation, we may be able to go a little further, for we would be 
able to guess with some accuracy the results of computations 
based on the data.) But some such problem as whether No. 2 



26 



CURT ROSENOW 



has some characteristic peculiar to it, or whether it is exhaustively 
represented by the other fourteen, would be quite insoluble. 

To answer such a question we must resort to partial correla- 
tion. Table 2 shows the effect of successively eliminating the 
effect of each variable, as we may conveniently call our tests, on 
the relation between No. 2 and No. 1. 

Table I 





1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 


I 

2 


28 






























3 


17 


44 




























4 


17 


-19 


-01 


























5 


10 


06 


-20 


55 
























6 


26 


05 


14 


-10 


-07 






















7 


25 


30 


04 


08 


02 


13 




















8 


21 


03 


08 


00 


-14 


3i 


17 


















9 


-12 


10 


00 


02 


02 


08 


-03 


25 
















10 


11 


10 


05 


04 


-10 


-06 


2.5 


36 


17 














11 


10 


41 


04 


06 


09 


10 


17 


23 


14 


01 












12 


07 


13 


22 


00 


-01 


09 


22 


01 


-17 


16 


-09 










13 


09 


17 


18 


09 


06 


20 


11 


25 


08 


23 


20 


18 








14 


-03 


02 


-08 


-07 


-10 


11 


-04 


17 


15 


-18 


18 


-01 


02 






15 


08 


03 


-04 


-06 


-10 


18 


-04 


40 


!5 


04 


07 


01 


03 


12 




16 


05 


20 


14 


10 


03 


03 


25 


42 


09 


-04 


21 


13 


10 


16 


-01 



If the reader will now recall what was said about the associa- 
tion between "Residuals," the meaning of this table should be 
clear. For example r 12 . 16 = + 0.28 is a measure of the as- 
sociation of No. 1 and No. 2 in so far as neither No. 1 nor No. 2 











Table 2 


1-12 






= 


+ 0.28 


Tl2 


16 




= 


+ 0.28 


1-12 


15 16 




= 


+ 0.27 


I"l2 


14 15 


10 


= 


+ 0.27 


l"l2 


13 14 


• 16 


= 


+ 0.26 


Tl2 


12 13 


• 16 


== 


+ 0.26 


Tl2 


11 12 


• 16 


= 


+ 0.25 


ris 


10 • • 


• 16 


== 


+ 0.24 


r 12 


9 •• 


• 16 


= 


-J- 0.26 


r^ 


8 •• 


• 16 


= 


+ 0.31 


1-12 


7 • • 


• 16 


= 


+ 0.28 


Tl2 


6 • • 


• 16 


= 


+ 0.28 


fl2 


5 • • 


• 16 


= 


+ 0.28 


rja 


4 • • 


• 16 


= 


+ 0.36 


I-X2 


3 • • 


• 16 


= 


+ 0.32 



THE ANALYSIS OF MENTAL FUNCTIONS 27 

is associated with No. 16, (accuracy, 'Hard Directions'). But 
this value is identical with the one obtained before X i6 was elimi- 
nated. It follows, even though r 2 16 is + 0.20, that the relation 
of No. 2 to No. 16 is of no significance with reference to the 
relation which exists between No. 1 and No. 2. The converse, 
however, is not true. For ri i 6 — 0.05 and r x 16 . 2 — o. There- 
fore the relation of No. 1 to No. 16 is entirely accounted for by 
that which No. 2 and No. 16 have in common. These conclusions 
are subject to the limitations imposed on us by the small number 
of our observations and by the assumption of linear regression. 
We have discussed the second, and we will presently discuss the 
quantitative expression of the first of these limitations. 

Similarly r i2 . 34 .... i 6 = + 0.32 is a measure of the rela- 
tion of No. 1 and No. 2 in so far as both No. 1 and No. 2 are not 
associated with any of the other fourteen variables. This rela- 
tion is peculiar to academic marks and to Immediate Logical 
Memory (Auditory), alone. This answers the question we pro- 
pounded, and is analysis in a very real sense. But before we en- 
deavor to push the analysis to its logical conclusion and try to 
ascertain the character of this elementary function which we have 
isolated, we must turn our attention to the disillusionizing sub- 
ject of validity. We have deliberately avoided this topic so far, 
because its discussion would have added nothing to our under- 
standing of the causal reasoning which underlies the theory of 
correlation. Now that we have reached concrete results it can 
no longer be postponed. 

If a number of samples be taken from a universe of discourse 
{viz., the universe of college freshmen of America), and con- 
stants such as the Mean, the Standard Deviation, and the Coeffi- 
cient of Correlation be ascertained, the results will differ from 
what would have been obtained if the entire "uni verse" had 
served as a basis. Such deviations from the theoretical true 
value are called errors of sampling. Theoretically it is possible 
.to ascertain the probability that any given error of sampling 
falls within given limits. Practically, in the case of r, this is 
done on the basis of the assumption that distributon is normal. 
The conventional expression given to facilitate the computation 



28 CURT ROSENOW 

of this probability is the so-called probable error. In and of 
itself, however, it merely means that the deviation of r from its 
"true" value is as likely to be greater than the true value, as 
it is to be smaller. If a given r is equal to its probable error, the 
chances are i : i that it would arise by chance even though the 
"true" value be zero. If r = 2 P.E. (probable error), these 
chances are 1 : 5, roughly. For r = 3 P.E., they are 1 : 23, for 
r = 4 P.E., 1 : 143, etc. 1 — . When r = 3 P.E., r is said to be 
"significant." Of course, such a standard is conventional and 
arbitrary, and not all authorities recommend the same ratio. In 
any case it is well to bear in mind the meaning of this "signifi- 
cance." Wth these considerations in mind let us now return to 
the consideration of our data and results. 

In table 3 column 1 (see below) are given the r's of marks 
with each one of our tests, followed by the probable error of 
r. They are the coefficients of zero order. In column 2 the 
same values and their probable errors are given after the in- 
fluence of the other fourteen has been eliminated by partial cor- 
relation. These are the coefficients of the 14th order. They rep- 
resent the correlation with marks of what is unique to each test. 
Let us return to test 2 and 3 the logical memory test. r 12 = 
-f 0.28, r 13 = + 0.17. Does this show that "Auditory Presen- 
tation" is more highly correlated with marks than Visual Pre- 
sentation ? Not at all, for the difference is o. 1 1 and the probable 
error of this difference is 0.094. So the chances are about even 

Table No. 3 

2. Log. Mem. Aud. + 0.28 ± 0.065 + °-32 ± 0.063 

3. Log. Mem. Vis. + 0.17 ± 0.068 + 0.04 ± 0.070 

4. Loss or Gain in No. 2 + 0.17 ± 0.068 + 0.26 ± 0.066 

5. Loss or Gain in No. 3 + °- 10 ± °-°7° + °-°3 — 0.070 

6. Sentences Built + 0.26 ± 0.066 -f- 0.23 ± 0.067 

7. H. Directions, Speed + 0.25 ± 0.066 + 0.09 ± 0.070 

8. Con. Increment, Speed -}- 0.21 ± 0.067 + 0.21 ± 0.067 

9. Objects Seen — 0.12 ± 0.069 — 0.23 ± 0.067 

10. Number-Checking + 0.11 ± 0.069 ± 0.01 ± 0.070 

11. Opposites, Speed + 0.10 ± 0.070 + 0.12 ± 0.069 

12. Numbers Hard -f- 0.07 ± 0.070 ± 0.04 ± 0.070 

13. Words Built + 0.09 ± 0.070 ± 0.07 ± 0.070 

14. Opposites, Accuracy — 0.03 ± 0.070 ± 0.01 ± 0.070 

15. Con. Increment, Accuracy + 0.08 ± 0.070 ± 0.03 ± 0.070 

16. H. Directions Accuracy + 0.05 ± 0.070 — 0.14 ± 0.069 



THE ANALYSIS OF MENTAL FUNCTIONS 29 

that the difference is due to chance. But if we turn to the cor- 
responding r's of the 14th order, we find that r 12 . 3 4 . . . . 16 = 
-f- 0.32 and r 13 . 2 4 • • • 16 = + 0.04. The difference is 0.28, the 
probable error of the difference 0.094. Hence the chances that 
this difference is due to fluctuation of sampling alone are 1 : 23. 
By conventional standards, there is a valid difference between 
Auditory and Visual presentation. Now we can interpret. The 
difference is due to something that has not been eliminated. I 
can think of but three possibilities. They are, ( 1 ) a difference 
in subject matter, (2) speed of presentation. In the visual pre- 
sentation the subject reads at his own rate and may violate the 
instructions against re-reading. In auditory presentation he 
must accept the rate of speed of the experimenter. (3) A dif- 
ference specific to the sense avenue, possibly in conjunction with 
the previous experience of the individual. Our data do not per- 
mit of a choice between these three possibilities, but it would be 
a simple matter so to control these conditions in another series 
of tests that interpretation would be narrowed down practically 
to a single possibility. Of course someone else might think of 
other possibilities. But he should remember that the cause of the 
difference cannot be anything involved in any of the other tests, 
unless it be a very marked difference of degree, and also that 
elimination has been from the criterion as well as from the test. 
(I have not thought it necessary to mention possibilities which 
would come under the head of obvious control of conditions 
common to all careful experimentation.) 

Again, r 12 = + 0.28, r i4 == + 0.17. Recalling that 4 is the 
difference between the score in test No. 2 and the score made after 
two weeks, we may note that the factor of "immediate memory" 
has been eliminated not by partial correlation, but by the method 
of scoring. The change is so radical that it does not seem profit- 
able to compare 2 and 4 as modifications of the same test. We 
may note, however, that r is not significant by conventional 
standards, but becomes so as a partial of the 14th order, for 
r i4 • 23 • • • . 16 = + 0.26. Although it is not apparent from 
table 3 alone, the same thing is true of tests No. 3 and No. 5. No. 
3 and No. 5 are identical with No. 2 and No. 4 respectively ex- 



30 CURT ROSEN 0W 

cept as to subject matter and mode of presentation. Their cor- 
responding values, of zero order, are r 13 = + 0.17 and r 15 = + 
0.10. Their coefficients of the 14th order are 0.04 and 0.03. But 
we have already seen that Auditory presentation contains every- 
thing of significance in Visual presentation, so that there is 
nothing left to compare after No. 2 and No. 4 have been elimi- 
nated, as they have been in the 14th order. To make the com- 
parison we must contrast the coefficients of the 12th order. We 
find r 13 . 5 e . • 16 = + 0.19 and r 15 . 3 e • • • ie = + 0.20, results 
which are quite similar in character to what we found in the 
case of No. 2 and No. 4. 

Now the significance of No. 4 (and No. 5) has been covered 
up in some way. Consulting a table of some 1300 partial co- 
efficients which were computed in order to get our results, but 
which are not published, we find that r 14 . 2 = -\- 0.24. The 
same fact might have been guessed from r 24 = — 0.19 (see table 
No. 1). The significance of No. 4 then, is obscured by the 
fact that subjects who make a high score "immediately" tend to 
forget more than those who do not. No doubt this is at least 
in part due to the fact that they have more to forget. Of course, 
different results as to r 24 might have been obtained if the loss 
or gain had been expressed in percentage terms. (E.g., if X = 
80, X after two weeks = 60, then X — 20/80 = 25%.) But 
this would have been as arbitrary as the method chosen. Be- 
sides, and the fact is interesting, using partial correlation made 
us relatively independent in the matter of selecting a unit of 
measurement, for if we had adopted a percents" and had ob- 
tained say r 24 = o (instead of — 0.19), we would also have 
riad a different value for r 14 and would have had the value of 
r 14 . 2 = + 0.24, as before, unless other factors entered. 

Returning now to the interpretation of r 14 . 23 . . . . 16 , we have 
seen that it is significant and that it is a factor in r 12 which not 
only was itself hidden, but also operated to lower the value of 
r 12 . This factor cannot be subject matter, for that is the same 
in both cases. It cannot be "mode of presentation," for X 4 is not 
stated with reference to the material as presented, but with re- 
spect to the material as "immediately" retained. Besides the 



THE ANALYSIS OF MENTAL FUNCTIONS 31 

original mode of presentation is the same in both cases. The 
factor may be that of interest in the tests themselves. Inter- 
ested subjects will tend to rehearse their performances and will 
be likely to discuss the tests with others and to compare notes. 
Or else it is reasonable to argue that the subjects who have 
seized the essential significance of the passage will have an ad- 
vantage in the matter of permanent retention over those who 
depend on a more verbal type of memory, whereas in test No. 2 
as scored, only two units out of a possible 97 are allowed for 
the reproduction of the gist of the passage, and 54 units are 
allowed for the reproduction of "ideas" which may be nothing 
but words. 23 Again I am obliged to say that our data do not 
justify me in saying more. But it is obvious that a comparison 
of different methods of scoring would be likely to give a defi- 
nite conclusion to our problem. 

Let us now consider No. 8, the Constant Increment test. We 
note that r ls = + 0.21 and r 18 . 23 . . 16 = + 0.21. Both r's 
are significant, but it is difficult to draw a conclusion beyond 
the restatement of the fact that No. 8 contains a significant 
factor not contained by the other fourteen. As we shall see, 
even the 1300 coefficients referred to above do not contain the 
necessary information which would enable us to push analysis 
much further. But it is quite unthinkable that such a complex 
function as adding a constant increment under test conditions 
should resist analysis. Indeed analysis is possible on the same 
lines we have pursued so far. But now the amount of arith- 
metic necessary for analysis might easily become unthinkable. 
It is at this point that the method devised by the writer will 
enable us to proceed. 

Table No. 4, given below, is somewhat analogous to table No. 2. 
It shows the effect on the relation of marks to No. 8 of the 
elimination of each of the other fourteen tests for some one 
order. It is unlike table No. 2 in that elimination is not consis- 
tently successive or continuous. The reason for this is practical. 
To have the order of elimination successive and continuous for 

23 Kitson, op. cit.j p. 25. 



32 CURT ROSENOW 







Table No. 4 


r« 




== + 0.21 


ris 


16 


= + 0.21 


1-18 


15 16 


= + 0.19 


r 18 


14 15 16 


= + 0.20 


r 18 


13 16 


= + 0.18 


r 18 


12> • • -16 


= + 0.19 


r« 


11 • • • «16 


= + 0.18 


ris 


• 10 • • • • 16 


= + 0.17 


r 18 


. 9 • • • • 16 


= + 0.18 


r 18 


• 2 


— + 0.21 


r« 


•2 3 


P + 0.21 


r 18 


•234 


= + 0.21 


r 18 


•2345 


= + 0.21 


ris 


• 2 • • • • 6 


= + 0.13 


r i8 


• 2 • • • • 7 


= + 0.12 



each of the fifteen correlations of type r, would involve the com- 
putation of 8400 coefficients, instead of the 1300 which we have. 
(See appendix.) We note that r 18 fluctuates about its origi- 
nal value except at the point where No. 6 (Sentences Built) is 
eliminated, and there it drops to + 0.13. I am unable to offer 
any very convincing interpretation as to the factor which is alike 
in these two tests. Instead we will face the question how 
ris • 2 • • • • 16 regains, so to speak, its original value. Clearly 
table No. 4 does not furnish an answer. Neither do any of the 
1300 coefficients from which it is an excerpt. The reason is as 
yet hidden somewhere in the infinite complexities of a situation 
which involves sixteen variables. Our only hope of finding this 
factor, without an inordinate amount of arithmetic, lies in re- 
ducing the number of variables. Now the factor, be it what it 
may, must be significantly related to marks. Therefore, if we 
can find the variables which, when combined, include every- 
thing which is so related, and if we can exclude those which 
merely duplicate some factor or set of factors, we will be that 
much nearer to the solution of the problem. 

Let us recall that R, the coefficient of Multiple Correlation, is 
a measure of the relation of a number of combined variables to 

another variable. Ri( 2 3 4 ie), the relation of our fifteen 

tests to marks, is + 0.55. If now we combine the five variables 
having the highest coefficients of the 14th order (see table No. 



THE ANALYSIS OF MENTAL FUNCTIONS 33 

3 col. No. 2) we find by computation that Ri( 2 4 e s 9) = + 0.52. 
The difference between the two R's is 0.03, probable error 0.09 
(roughly), and therefore not only is not significant, but the 
chances are 5:1m favor of its being due to chance. These 
five variables therefore contain everything that is significantly 
related to Marks, including the factor we are looking for. So 
if we eliminate successively No. 2 (Log. Mem. Aud.), No. 4 
(Loss or Gain in No. 2), No. 6 (Sentences Built), No. 8 
(Const. Increment), and No. 9 (Objects Seen), from r 18 our 
problem will be solved. 

Table No. 5 contains these values (see below). Inspection of 

it shows at once that r i8 drops, as before, when No. 6 is 

eliminated, but rises again when No. 9 is eliminated. The fac- 
tor we are looking for is common to No. 8 and No. 9, and is 
not in No. 2, No. 4, or No. 6. 

Table No. 5 
r i8 = + 0.21 

1-18 . 2 = + 0.21 

ri 8 • 2 4 = + 0-21 

r 18 • 2 4 e = :f 0.14 
r 18 . 2 4 e 9 = + 0.19 

What is this factor? Well, I regret to say that we may save 
ourselves the trouble of interpretation, for the difference we 
have investigated is not based on a sufficient number of obser- 
vations to be "significant." I have isolated this "factor" in 
order to exhibit what appears to me as the beauty of the analy- 
sis and in order to illustrate what can be done by means of in- 
direct analysis, (analysis by exclusion) by means of the manipu- 
lation of R, a method, which, so far as I know, has not been 
suggested elsewhere. 

Similar remarks are in order for tests No. 6 and No. 9. In 
both cases the correlation of the 14th order is significant. In 
both cases we have a fairly high degree of probability that each 
test possesses something, peculiar to it alone, significantly cor- 
related with marks. Beyond this point our 92 observations 
will not permit us to go statistically. We may go further, if 
we like, by way of a priori reasoning which has perhaps sug- 



34 CURT ROSEN OW 

gestive value. Moreover this value is enhanced by the fact that 
the elimination of fourteen fairly diverse tests limits us to sug- 
gestions which must be fairly concrete and — which is more to 
the point — verifiable by future investigations. Before the close 
of this paper I shall be guilty of a little speculation of this kind. 

So far we have been occupied exclusively with analysis. To 
be sure, the paucity of our data, combined with the low value of 
our correlations, did not permit us to go very far. But I trust 
that the sort of thing which might be attained with more ex- 
tensive data is clear. We may now turn to the subject of diag- 
nosis and prognosis. 

We have seen just now that five tests carry practically all the 
meaning, with reference to marks, contained in the fifteen tests. 
It follows that they carry also all the diagnostic value. We 
have shown in an earlier part of this paper that the value of R 
which may arise owing to fluctuations of sampling alone may 
easily become unpleasantly large. We found that for 16 vari- 
ables (15 tests and a criterion) and 92 observations the "prob- 
able" value of R is + 0.40 even when none of the tests have 
any actual relation to the criterion or to each other. Our result 
of -\- 0.55, compared to this, has but little significance, and this 
is in itself a sufficient reason why it has little value for diag- 
nosis. (I dare not say no value, on account of the argument 
we hear so often that an infinitesimal part of a loaf is better 
than no bread at all.) The case is more favorable if we combine 
our five "best" tests. Here the actual R is + 0.52, and the 
corresponding "chance" value is -f- 0.21. The difference, at 
any rate, is significant, and, if anyone cares to do it, he may 
compute the weights which these five tests have in their regression 
equation and use the equation for the "practical" diagnosis of 
the ability of individual students. 24 But considerations of a 
more familiar sort show us the trivial nature of the diagnostic 
value, even in this case, in still more drastic fashion. Let us 
assume that our R = + 0.55 not only is significant, but is 
absolutely correct for the entire "population" of freshmen in 
America. What then would be its diagnostic value? The 

2 *A short method for computing weights wall be found in the appendix. 



THE ANALYSIS OF MENTAL FUNCTIONS 35 

probable error of R is now zero, but the "Standard Error of 
Estimate" remains. In discussing the logic of partial correla- 
tion we brought out the significance of this expression as being 
a "Residual." Now such a Residual, if it remains on our hands 
after we have estimated on the basis of all the information at 
hand, — be that information supplied by one or by fifty tests, — 
such a Residual is an Error of Estimate due to unknown causes. 
The standard deviation of all such errors is therefore a meas- 
ure of the accuracy attainable on the basis of such information. 
In terms of the standard deviation of the variable we are esti- 



mating (say academic marks), it is V 1 — r2 - As r approaches 
zero, V i — r 2 approaches I, Or, in other words, all of our 
estimates tend to coincide at the mean. Or, in still other words, 
in the absence of relation our average error of estimate will be 
least if we estimate all deviations as zero. Now when r = 0.55, 
^ 1 — r e = 0.84. The gain in diagnostic value is thus very 
slight. 25 

The case now stands as follows. The values of R, which we 
found, had no significance for a combination of fifteen tests,, 
and only a moderate significance for a combination of five tests. 
But even if these values were absolutely correct, their diagnostic 
value would be very low. On the other hand our analysis of the 
Logical Memory test retains such significance as we showed it 
to have and, even though the lack of material did not permit of 
pushing the statistical analysis very far, it has focussed our at- 
tention on five of the tests in a way which could not have been 
anticipated from the raw material, and has, in some measure, 
cleared the way for further investigations. 

Is this then the last word that can be said for "practical" diag- 
nosis? By no means. To be sure, if diagnosis of individual abil- 
ity be the end in view, the material we have investigated is 
worthless. Furthermore, I believe that this is true of all similar 
work done by others, in so far as their results can be duplicated 

25 The formula for the error of estimate is, of course, perfectly familiar. 
But, whatever may be true of others, I had never realized how very rapidly 
it increases in value as r becomes less than unity. My attention was called 
to the fact by Mr. B. Ruml. 



36 CURT ROSEN OW 

by other investigators. But then, if diagnosis of individual abil- 
ity for the purpose of educational guidance of the individual 
student were the only use to which tests and correlations could 
be put, I would not give the subject five seconds of my time. The 
motivation of this position would lead us too far afield, and we 
may waive discussion. But, the one who really needs guidance 
is the educator. And, even if the reader cannot agree with the 
first point, he will surely assent to the second. Now if it could 
be shown that a purely verbal type of memory has a correlation 
of zero with academic achievement at one institution or in one 
department, 25 at another, 50 at still another, etc., it would throw 
very little light on the "General Intelligence" of any individual 
student, but it would furnish a world of "guidance" to the edu- 
cator. For, assuming that it is undesirable to encourage and 
develop the "verbal" type, he would be able to direct his atten- 
tion and energy to the task of making other institutions or de- 
partments conform to the standard set by the first. It is to facts 
in the mass that statistics properly applies. It is there it can and 
should be applied. 

IV. Conclusions based on the Data 

It remains to regale the reader with the a priori speculation 
with which he has been threatened. We may conveniently do 
this by casting it into the form of "conclusions." At the same 
time we will restate the more valid conclusions which have re- 
sulted from the discussion. 

( 1 ) The whole collection of tests has a low diagnostic value. 

(2) This is due only in part to the low value of the individual 
correlations. It is due very largely to the enormous amount of 
duplication. The fifteen tests are, to a large extent, all measures 
the same thing or things. This fact enables us to concentrate 
our attention on five tests. 

(Hereafter, in discussing the correlations of the various tests, 
reference will be to the coefficients of the highest order, unless 
specifically stated otherwise.) 

(3) The Logical Memory Test: The probability is 1300: 1 
that test No. 2 is significant. The probability is 23 : 1 that audi- 



THE ANALYSIS OF MENTAL FUNCTIONS 37 

tory presentation is "superior" to visual presentation. We need 
not repeat the discussion. We must not however interpret it to 
the effect that the "visual" test is of no value. It becomes super- 
fluous when auditory presentation is used. 

The probability is 140: i that test No. 4 (loss or gain) is sig- 
nificant. This is a clue toward the differentiation of different 
types of memory. Comparison of various modes of scoring sug- 
gests itself. E.g., we may compare, in No. 2, an evaluation based 
simply on the number of words correctly repeated with other 
methods. The significance of No. 4 may also be due to the factor 
of "interest." To a lesser degree of probability, similar state- 
ments can be made about test No. 5. 

(4) The probability is about 25 : 1 that test No. 8 (Const. In- 
crement) is significant. The instructions for this test empha- 
size speed more than those for any of the others where speed is 
a factor at all. Moreover the activity is far more mechanical and 
automatic, approximating simple reaction time after practice. 
Indeed for the faster subjects the limit of speed seems to be 
physiological rather than mental. Even some subjects who are 
only fairly fast give this impression, i.e., it looks as if they cannot 
talk as fast as they can add. This, physiological reaction time, 
may be the significant factor. It might be investigated by using 
1 as the increment, adding only to single digits, and reducing 
the number to fifty. On the other hand the test, as given, is 
monotonous and long. It is hard work, and fast subjects are 
a little out of breath when they finish. As has been said, the 
raw data were not available, but I have noted elsewhere that some 
subjects increase their speed as they go along and spurt at the 
end, while others become slower and slower. By contrasting 
speed say in four quarters we might get a measure of some of 
the so-called character qualities, such as effort, preseverance, etc. 
Some such thing may be the significant factor in the Constant 
Increment Test. Of course, all this is highly speculative. 

(5) The probability is about 30: 1 that test No. 9 (Objects 
Seen) has negative significance. Of the ten objects presented, 
the fountain-pen, penfiller and inkwell, the envelope and the two 
cent stamp, the pencil and the ruler, are fairly well associated. 



38 CURT ROSEN OW 

The association for the 25 cent piece, the maroon ribbon, and the 
key are more remote. The mean for all subjects was a little over 
seven, so that those who recalled more than the number of objects 
which are closely associated tended to have low marks. Now it 
is well known that the capacity to note and to retain a mass of 
irrelevant details is very great in the hypnotic trance, and many 
authorities incline to the belief that the trance is simply a state 
of diffuse attention. It might very well be that a selective, dis- 
criminating type of mind will obtain low scores in this test. The 
suggestion is of course capable of experimental investigation by 
varying the nature of the objects. 

(6) There is a probability of about 30: 1 that test No. 6 
(Sentences Built) is significant. I have no comment to offer. 

(7) Accuracy appears to be of no significance except possibly 
as it enters into the logical memory tests. In the case No. 7 
(Hard Directions) there is even a probability of 5 : 1 that it has 
negative significance. Now it is fairly well established that 
speed and accuracy are positively correlated at many activities 
and this is also born out by our data. Thus r 8 15 — + 0.40 
(Const. Inc.), r 7 16 = + 0.25 (H. Directions) and r lx 14 = -j- 
0.18 (Opposites). On the other hand the relation is probably in- 
verse, within limits, for any given individual. Of course, our 
tests cannot show the deviation of an individual from his own 
point of maximum efficiency. So it would seem that in so far as 
speed is significant, it has associated with it all of the significance 
of accuracy. We have already seen that in, so far as speed is 
significant, it seems to be most adequately represented by the 
Constant Increment test. 

If it is hard to say why a test does correlate, it is even more 
difficult to indicate why it does not. We may say in a general 
way that the ten tests whose coefficients of the highest order give 
no indication of significance, are very likely influenced by gen- 
eral factors, such as ability to adapt to novel situations such as 
the entire test situation is for the average subject, by interest, 
effort, etc. We can only say that in so far as they have value 
they are more adequately represented by the five tests with the 
highest coefficients of the 14th order. 



THE ANALYSIS OF MENTAL FUNCTIONS 39 

This is all that I find it useful to say of the specific results ob- 
tained in the present study. We will now turn to an exposition 
of the mechanical technique involved in analyzing a complex 
situation. 

V. Appendix : A Contribution of the Mechanical Tech- 
nique of Partial Correlation 

In what follows it is assumed that the reader already has a 
working- knowledg of the theory of correlation. To proceed on 
any other assumption would be impossible in a work of the scope 
of the present one. On the other hand it has been my aim to 
arrange the steps in such a way that any-one who understands 
the terminology can follow the steps mechanically, or at least by 
symmetry, without having to inquire about the why's and where- 
fore's. 

The general problem before us is this: Given a dependent 
variable and a number of independent ones, how can we obtain a 
maximum of information with a minimum of arithmetic. In so 
far as the schema devised by me is applicable to such a problem 
it can be divided into three sub-problems, which however form 
successive stages of what really is a single operation. These 
three stages are: (i) The finding of the coefficient of Multi- 
ple correlation. (2) The finding of the coefficients of partial 
correlation, with reference to the dependent variable, of the high- 
est order. (3) The finding of the coefficients of regression 
(weights) for estimating the dependent variable. 

For the sake of simplicity we will assume that all standard de- 
viations of zero order are 1. Also, in order to avoid the cum- 
bersome use of "n," let us suppose that we are dealing with five 
independent variables, and one dependent one. The method can 
be extended to any number whatever by symmetry. 

Problem I 

Find Ri( 23456 )• This may be done directly from the equation 26 

1— R 2 =(1— r 2 )(i— r 2 )(i-r 2 )(i-r 2 )(i— r 2 ) No. 1 

1(23456) 12 13.2 14.23 15.234 16.2345 

Beginning with the raw data, the work proceeds as follows : 

(1) Find all coefficients of zero order. 

26 Yule. Intro., p. 248. 



40 CURT R0SEN0W 

(2) Write equation No. 1 as above, and rewrite it, reversing the 

order of subscripts as follows : 

1— R» =(1— r 2 )(i-r 2 )(i— r 2 )(i— r a )(i— r 2 ) No. 2 

1(23456) 16 15.6 14.56 13.456 12.3456 

(3) Compute the coefficients of equation No. 1 and No. 2. For 
this purpose there will be needed all coefficients having the 
secondary subscripts in these two equations, forty all told. 
That is, for equation No. 1, we will need all coefficients of 
type r_. 2 , r_. 23 , r_. 23 4, r_. 2345 . It is convenient to prepare a 
list of these coefficients on which their values can be entered 
as fast as computed. Such a list can be written by sym- 
metry with a minimum of thought, and about as fast as one 
can write. Giving subscripts only, it is : 

Table I 
132 15.6 

14.2 14.23 
15.2 15.23 15.234 
I16.2 16.23 16.234 16.2345 12.6 12.56 12.456 12.3456 

34.2 
35-2 
36.2 

45-2 45-23 
46.2 46.23 

56.2 56.23 56.234 

(4) Compute 1 — R 2 according to equations No. 1 and No. 2. 
The two values should check. They afford an independent 
check on all of the previous arithmetic with exception of 
the computation of the coefficients of zero order. 

(5) Compute R, or, preferably, look it up. 27 

Problem II 
To find the coefficients of the fourth order with reference to 
the dependent variable. There are 5 such coefficients. They are : 

ri2'3456 
ri3-2456 
ri4-2356 
ri5-2346 
ri6-2345 

27 Use "Tables for statisticians and Biometricians." K. Pearson, Table 8, 
pp. 20-21. 



14.6 

13.6 

12.6 


14.56 
13-56 
12.56 


13456 
12.456 


25.6 
24.6 
23.6 


24.56 
23.56 


23456 


35.6 

34.6 


3456 




45.6 







We may write, 



THE ANALYSIS OF MENTAL FUNCTIONS 41 

1*12.3456 and 1*16.2345 have been found in Problem No. 1. The other 
three can be found indirectly from the following consideration. 
Let Ri(_a) be the Multiple coefficient showing the relation of X x 
to a combination of all the independent variables except X a , e.g., 
Ri(- 2 ) is Ri( 3456 ). 

No. 3 

,, I R 2 l(abc-n) 2 

then P2 ; = 1— r 2 i a .bc__n 

I — XV. l^-aj 
a3 _ . I R 2 l(23456) a 

so that _». ; H 1 = 1 — r 2 i8.2 4se 

I— R 2 l(2456> 

1 — R 2 1(23456) , 

■ ^ — 7 r- — I r 14.2356 

1 — R 2 1(2356) 

1 — R 2 1(23456) „ 

; ^y~7 T— = I r 15.2346 

I K l(2346) 

i— R» =(1— r 2 )(i-r 2 )(i-r 2 )(i— r l ) 

1(2456) 16 15.6 14-56 12.456 

1— R' =(1— r 2 )(i— r 2 " )(i-r 2 )(i— r 2 ) 

1(2356) 16 15.6 13.56 12.356 

1— R 2 =(1— r 2 )(i— r 2 )(i-r 2 )(i— r 2 ) 

1(2346) 12 13.2 14.23 16.234 

Now all of these coefficients are in table I with exception of ri 2 . 35 6, 
which may be computed from r 12 . 5 6, ri 3 . 56 , and r 23 .56 which also 
are in table I. We are now able to compute all of the coefficients 
of the fourth order from equations of type No. 3. 

It will be noted that coefficients found indirectly from equation 
No. 3 are indeterminate as to sign. However, unless the numeri- 
cal magnitude of such coefficients is negligible, it will usually be 
possible to determine the sign by inspection. 

Problem III 
Find the coefficients of regression. The familiar expression 

<"l.(2 3 4. -n) 

for this coefficient is, b12.34.-n = 1*12.34. . n 

°2.1 3 4- -n 

We need therefore, in addition to the correlation coefficients of 
the fourth order, all of the six standard deviations of the fifth 
order. These should be written as follows : 



42 



CURT ROSEN OW 



O"! -23456 : 



C2-13456 : 



ffVl2456 : 



0"4-l2356 : 



(T5-12346" 



0"6-12345 : 



[' 



I 



-r 2 )(i 

16 



-r 2 )(i-r 2 )(i- 
15.6 14.56 



)(i- 



36 

1-r 2 )(i- 
46 



25.6 



)(i-r 2 



24.56 



)(!■ 



35-6 



)(i-r 2 



34-56 



)(i- 



13456 

)(i 
23456 

■1- )(i- 

23.456 



)(i-r 2 



12.3456 

) 
12.3456 



r 
r 



45-6 



) (l _r. ) (l _ r * 



34.56 



)(i-r 2 



24.356 



) 
13.2456 

) 
14.2356 



X 



Y 



1-r 2 )(i-r 2 )(i-r 2 )(i- 

25 35-2 45-23 

-r 2 )(i-r 2 )(i— r 2 )(i— 1 

26 36.2 46.23 



r 2 )(i-r 2 

56.234 



)1 

15-2346 J 



H 



56.234 



)(i-i 



) 

16.2345 



r 



The only new coefficient in all of the above expressions is r 24 .356. 
It can be found from r 23 . 5 6, r 24 . 56 , and r 34 . 5 6, table I. The above 
six expressions for the standard deviations of the fifth order are 
written in such a way that the dependent variable will enter into 
the last term as a primary subscript, and from this point the 
variables are eliminated as nearly as possible in the same order 
in which they have been eliminated in equations No. 1 and No. 2, 
choosing by inspection the equation which is seen to be best for 
the purpose. 

From this point on the computation of the regression coeffi- 
cient proceeds in the usual way. 

The method outlined will, of course, lend itself to the solution 
of a variety of problems. The only point at all novel is expressed 
by equation No. 3, problem No. 2. In spite of its very great 
simplicity, I have never seen it in print, probably because it has 
no theoretical interest. It does however afford a very useful 
shortcut to the arithmetic, and the rest of the schema is simply a 
systematic exploitation of this fact. 

In comparing this schema with the one given in Yule's "Intro- 
duction," it should be borne in mind that Yule's schema provides 
for the finding of all possible relations. So far as I know, there 
is no short cut to this problem. I have simply taken advantage 
of the fact that in "test" work interest centers, or should center, 
on one or two dependent variables, not themselves tests. 



THE ANALYSIS OF MENTAL FUNCTIONS 43 

The schema will show to advantage, however, if contrasted 
against's Kelley's. 28 Kelley faces practically the same problem as 
I, except that he scarcely mentions analysis and puts the emphasis 
exclusively on diagnosis and prognosis. He states 29 that the 
number of coefficients of partial correlation to be computed is 2 
in the case of three variables, 15 in the case of 4 variables, 36 
in the case of 5 variables, 78 in the case of 6 variables, etc. Our 
schema requires 2 coefficients in the case of 3 variables, 8 in the 
case of 4 variables, 22 in the case of 5 variables, 45 in the case 
of 6 variables, etc. Moreover, in our method, the symmetry of 
table I, which accomplishes practically all of the work, is very 
easily seen and reproduced, and it requires only a very moderate 
amount of practice and ingenuity to write the expressions in 
problems No. 1 and No. 2 to the best advantage. Over against 
this, Kelley does not state any guiding principle for selecting the 
coefficients to be computed to the best advantage and I am quite 
unable to see the symmetry without such aid. For example I 
am quite unable to say how many coefficients Kelley would need 
for say 8 variables. But then I am quite prepared to find that 
that is due to my own stupidity. 

Finally I wish to call attention again to the very useful char- 
acter of Kelley's tables. Using these tables in conjunction with 
a "millionaire" calculating machine, 75 coefficients per hour, 
correct to two places, or 40 coefficients, correct to three places, 
can easily be computed. 

28 Op. cit., p. 23, this paper. 

29 Op. cit., p. 14. 



021 068 659 9 



